At the end of the course, each student should be able to…
- C01: SurfaceEQ. Identify and sketch surfaces in three-dimensional Euclidean space.
C01. Surfaces in Three-Dimensional Space
- Simple planes parallel to coordinate axes are given by \(x=a\), \(y=b\), and \(z=c\).
- The dot product may be used to obtain equations for arbitrary planes.
- The equation \(\vect N\cdot(P-P_0)=0\) describes the points \(P\) on the plane passing through the point \(P_0\) and normal to the vector \(\vect N\).
- If \(P=\<x,y,z\>\), \(P_0=\<x_0,y_0,z_0\>\), and \(\vect N=\<A,B,C\>\), then this equation may be expressed as \(A(x-x_0)+B(y-y_0)+C(z-z_0)=0\) and simplifies to \(Ax+By+Cz=D\) for some value of \(D\).
- Quadric surfaces are another common example used in calculus.
- These surfaces are obtained from quadratic equations (polynomials of degree \(2\)) of the variables \(x,y,z\).
- The simplest example is that of the sphere \(\|P-P_0\|=r\) describing the points \(P\) that are distance \(r\) from the sphere’s center \(P_0\).
- If \(P=\<x,y,z\>\), \(P_0=\<x_0,y_0,z_0\>\), and \(\vect N=\<A,B,C\>\), then this equation may be expressed as \((x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2\).
- The surfaces given by equations of \(x,y,z\) may be sketched by considering traces of the surface in simple planes \(x=a\), \(y=b\), or \(z=c\).
- University Calculus: Early Transcendentals (3rd Ed)
- 11.1 (exercises 1-30, 41)
- 11.5 (exercises 21-26)
- 11.6 (exercises 1-12, 33-44)